Before you start with one arm push-ups you should be able to do many (at least 30) regular push-ups easily. The one hand push-up requires strong triceps and shoulders but also balance and a strong core.
How to perform one arm push-ups
Start in regular push-up position and place on arm on your side. (putting your hand on your hamstrings will give you a little extra stability and body tension).
You will feel that the wider you spread your legs the easier it becomes to keep your balance. The challenge here is to keep your legs as close as possible of course. You could start with a (lighter) inclinde one arm push-up as shown on the right. Once your shoulders and triceps become stronger, you could move to a more horizontal position.
Watch the video tutorial for an example and more tips: